M3 January 2008 Q4
4. A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle 30\(^\circ\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2mg\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\tfrac{1}{4}a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\tfrac{1}{2}a\).
By using the principle of conservation of energy,
| Scheme | Marks |
|---|---|
| Energy equation with at least three terms, including K.E term | M1 |
| \(\dfrac{1}{2}mV^2 + ..\) \(+ ..\ \dfrac{1}{2}.\dfrac{2mg}{a}.\dfrac{a^2}{16},\ + mg.\dfrac{1}{2}a.\sin 30,\ = \dfrac{1}{2}.\dfrac{2mg}{a}.\dfrac{9a^2}{16}\) | A1, A1, A1 |
| \(\Rightarrow V = \sqrt{\dfrac{ga}{2}}\) | dM1 A1 |
| (6) |
Notes
In part (a)
DM1 requires EE, PE and KE to have been included in the energy equation.
If sign errors lead to \(V^2 = -\dfrac{ga}{2}\), the last two marks are M0 A0
In parts (a) and (b) A marks need to have the correct signs
| Scheme | Marks |
|---|---|
| Using point where velocity is zero and point where string becomes slack: | |
| \(\dfrac{1}{2}mw^2 = \dfrac{1}{2}.\dfrac{2mg}{a}.\dfrac{9a^2}{16},\ - mg.\dfrac{3a}{4}.\sin 30\) | A1, A1 |
| \(\Rightarrow w = \sqrt{\dfrac{3ag}{8}}\) | A1 |
| (4) | |
| (10 marks) |
Notes
In parts (a) and (b) A marks need to have the correct signs
In part (b) for M1 need one KE term in energy equation of at least 3 terms with distance \(\dfrac{3a}{4}\) to indicate first method, and two KE terms in energy equation of at least 4 terms with distance \(\dfrac{a}{4}\) to indicate second method.
Alternative (using point of projection and point where string becomes slack):
| \(\tfrac{1}{2}mw^2 - \tfrac{1}{2}mV_1^2,\ = \dfrac{mga}{16} - \dfrac{mga}{8}\) | M1,A1 A1 |
| So \(w = \sqrt{\dfrac{3ag}{8}}\) | A1 |
SHM approach in part (b). (Condone this method only if SHM is proved)
| Using \(v^2 = \omega^2\left(a^2 - x^2\right)\) with \(\omega^2 = \dfrac{2g}{a}\) and \(x = \pm\dfrac{a}{4}\). | M1 A1 A1 |
| Using ‘\(a\)’ \(= \tfrac{a}{2}\) to give \(w = \sqrt{\dfrac{3ag}{8}}\). | A1 |